Differentiate `sin^(-1)sqrt(x)(0ltxlt1)` from first principles.

Explanation

We have given: `sin^(-1)sqrt(x)`

`d/dx(f(x))` = `lim_(hto0)(f(x + h) -f(x))/(h)`

`d/dx(sin^(-1)sqrt(x))` = `lim_(hto0)(sin^(-1)sqrt(x+h)- sin^(-1)sqrt(x))/(h)`

`d/dx(sin^(-1)sqrt(x))` = `lim_(hto0)(sin^(-1)(sqrt(x+h)sqrt(1-x)-sqrtxsqrt(1-x-h)))/h`

Let `z=sqrt(x+h)sqrt(1-x)--sqrtxsqrt(1-x-h)`

As `h to 0`then `z=sqrtxsqrt(1-x)-sqrtxsqrt(1-x)`

`implies h to 0, z to 0`

`d/dx(sin^(-1)sqrt(x))` = `lim_(hto0)(sin^(-1)z)/zxx(z/h)`

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